## The Schaefer model and its extensions

In life, there are few things that ''everybody knows,'' but if you are going to hang around anybody who works on fisheries, you must know the Schaefer model, which is due to Milner B. Schaefer, and its limitations (Maunder 2002, 2003). The original paper is hard to find, and since we will not go into great detail about the history of this model, I encourage you to read Tim Smith's wonderful book (Smith 1994) about the history of fishery science before 1955 (and if you can afford it, I encourage you to buy it). The Schaefer model involves a single variable N(t) denoting the biomass of the stock, logistic growth of that biomass in the absence of harvest, and harvest proportional to abundance. We will use both continuous time (for analysis) and discrete time (for exercises) formulations:

If you feel a bit uncomfortable with the lower equation in (6.8) because you know from Chapter 2 that it is not an accurate translation of the upper equation, that is fine. We shall be very careful when using the discrete logistic equation and thinking of it only as an approximation to the continuous one. On the other hand, for temperate species with an annual reproductive cycle, the discrete version may be more appropriate.

The biological parameters are r and K; we know from Chapter 2 that, in the absence of fishing, the population size that maximizes the growth rate is K/2 and that the growth rate at this population size is rK/4. When these are thought of in the context of fisheries we refer to the former as the population size giving maximum net productivity (MNP) and the latter as maximum sustainable yield (MSY), because if we could maintain the stock precisely at K/2 and then harvest the biological production, we can sustain the maximized yield. That is, if we then maintained the stock at MNP, we would achieve MSY. Of course, we cannot do that and these days MSY is viewed more as an upper limit to harvest than a goal (see Connections).

Myers et al. (1997a) give the following data relating sea surface temperature (T) and r for a variety of cod Gadus morhua (Figure 6.4a; Myers et al. 1997b) stocks (each data point corresponds to a different spatial location). Construct a regression of r vs T. What explanation can you offer for the pattern? What implications are there for the management of ''cod stocks''? You might want to check out Sinclair and Swain (1996) for the implication of these kind of data.

 r (per year) T (°C) r (per year) T (°C)
0 0